1simplices

A 1-simplex is the simplest nontrivial simplex. In Euclidean space it is the convex hull of two distinct points, which is geometrically a closed line segment. In an abstract simplicial complex, a 1-simplex is represented by an unordered pair of distinct vertices, often written as {v0, v1}; when orientation is required, it is denoted [v0, v1], giving a directed edge from v0 to v1.

The standard model of a 1-simplex is the 1-simplex Delta^1, defined as the set of points (t0, t1) in R^2 with t0, t1 >= 0 and t0 + t1 = 1. An affine map from Delta^1 into a topological space X, f: Delta^1 -> X, is called a 1-simplex in X. Geometrically this amounts to choosing an edge in X together with an orientation from one endpoint to the other.

Boundary and orientation are central in algebraic topology. The boundary of the oriented 1-simplex [v0, v1] is ∂[v0, v1] = [v1] − [v0], where [vi] denotes the corresponding 0-simplex (vertex). In the simplicial chain complex, C1 is the free abelian group generated by oriented 1-simplices, and the boundary map ∂1: C1 -> C0 records how edges connect to vertices.

Applications include the representation of edges in triangulations and networks, where 1-simplices serve as the edges linking vertices. They form the building blocks of higher-dimensional simplices in triangulations and play a key role in defining homology and other invariants via chain complexes. In simplicial sets and category theory, 1-simplices correspond to morphisms from the standard 1-simplex, encoding directed edges in combinatorial models.